Zeta functions of finite Schreier graphs and their zig-zag products

Abstract

We investigate the Ihara zeta functions of finite Schreier graphs [Formula: see text] of the Basilica group. We show that [Formula: see text] is two sheeted unramified normal covering of [Formula: see text], for all [Formula: see text] with Galois group [Formula: see text] In fact, for any [Formula: see text], the graph [Formula: see text] is [Formula: see text] sheeted unramified, non-normal covering of [Formula: see text]. In order to do this, we give the definition of the [Formula: see text] [Formula: see text] [Formula: see text] of Schreier graphs of Basilica groups. We also show the corresponding results in zig-zag product of Schreier graphs [Formula: see text] with a [Formula: see text]-cycle.